This is similar to the proof of Lemma REF . Using the twisted 4th moment formula with shifts in [1]} and lemma 2.5.1 of [2]}, we can write up to a power savings in \(T\)
\(I_T&(\alpha ,\beta ) = \int _{\mathbb {R}} \zeta (\tfrac{1}{2} + \alpha + i t)\zeta (\tfrac{1}{2} + i t)\zeta (\tfrac{1}{2} +\beta - i t)\zeta (\tfrac{1}{2} - i t) |A(\tfrac{1}{2} + i t)|^2 \phi (t/T) dt \\&= \frac{1}{4(2\pi i)^4} \oint \limits _{|z_j| = 3^j/\log T} A(z_1,z_2,-z_3,-z_4) G(z_1,z_2,-z_3,-z_4) \Delta (z_1,z_2,z_3,z_4)^2 \\&\quad \times \left(\int _\mathbb {R}\left(\frac{t}{2\pi }\right)^{\tfrac{z_1 + z_2 -z_3 - z_4 - \alpha - \beta }{2}} \phi (t/T) dt \right) \prod _{m=1}^4\frac{dz_m}{z_m^2 (z_m -\alpha )(z_m + \beta )},\) 

where \(\Delta (z_1,z_2,z_3,z_4) = \prod _{1\le j < k \le 4} (z_k-z_j)\)  is the Vandermonde determinant and
\(A(z_1,z_2,z_3,z_4) = \frac{\zeta (1+z_1 + z_3)\zeta (1+z_1 + z_4)\zeta (1+z_2 + z_3)\zeta (1+z_2 + z_4)}{\zeta (2+z_1 + z_2+ z_3 + z_4)}.\) 
Now differentiating with respect to \(\alpha \)  and \(\beta \)  and then setting \(\alpha = \beta = 0\)  gives after some algebraic manipulation
\(\widetilde{I}^{(2)}&(T)= \frac{1}{4(2\pi i)^4} \oint \limits _{|z_j| = 3^j/\log T} A(z_1,z_2,-z_3,-z_4) G(z_1,z_2,-z_3,-z_4) \Delta (z_1,z_2,z_3,z_4)^2 \\&\quad \times \Bigg [\int _\mathbb {R}\left(z_1^2 z_2^2 z_3^2 z_4^2 \left(\log \frac{t}{2\pi }\right)^2 - (z_1z_2z_3 + z_1z_2z_4 + z_1z_3z_4 + z_2z_3z_4)^2 \right)\\&\qquad \qquad \times \left(\frac{t}{2\pi }\right)^{\tfrac{z_1 + z_2 -z_3 - z_4}{2}} \phi (t/T) dt \Bigg ] \prod _{m=1}^4\frac{dz_m}{z_m^6}.\) 

Now to deduce the claim, notice that
\(A(z_1,z_2,-z_3,-z_4) \ll (\log T)^4, \quad \Delta (z_1,z_2,z_3,z_4)^2 \ll (\log T)^{-12},\) 
\(\int _\mathbb {R}\left(z_1^2 z_2^2 z_3^2 z_4^2 \left(\log \frac{t}{2\pi }\right)^2 - (z_1z_2z_3 + z_1z_2z_4 + z_1z_3z_4 + z_2z_3z_4)^2 \right) \\\times \left(\frac{t}{2\pi }\right)^{\tfrac{z_1 + z_2 -z_3 - z_4}{2}} \phi (t/T) dt \ll T (\log T)^{-6}\) 

for \(|z_j|= 3^j/\log T\)  and \(t\in [3T/4,9T/4]\) .
As in the previous proof, the analysis for the \(Z\)  function is simpler, and the same bound holds.
